You are referring to 2 Mongolias.
The State of Mongolia was formerly called the Mongolian People's Republic. Formerly a part of China, it proclaimed independence with the support of Russia. On March 13th 1921 a Provisional People's Government was established, and then on November 26th, 1924 the government proclaimed the Mongolian People's Republic. This was a Soviet satellite state.
A referendum took place on October 20, 1945, when Mongolia voted for independence.
After the breakdown of communist regimes in Europe in late 1989,
Mongolia saw its own democratic revolution in early 1990; it led to a
multi-party system, a new constitution of 1992, and transition to a
So what you are referring to in 1990 is the "modern" Mongolia. And the banks of modern state of Mongolia would be from 1990 onwards logically.
But coins have been issued by the "Mongolian People's Republic" since at least 1925. Example: KM#1, 9, 21 and 27 as per Catalogue each of denomination 1 MONGO and KM#2, 10 and 16 of denomination 2 MONGO. "State" (modern) Decimal Coinage seems to start from 1994 with KM#122 of denomination 20 TUGRIK. This seems to correspond with the idea of the new state and its Banks starting off around 1990.
Note that I referred to the 2008 catalogue rather than the 2015 linked above.